SS257a Midterm Exam Monday Oct 27 th 2008, 6:30-9:30 PM Talbot College 342 and 343. You may use simple, non-programmable scientific calculators.

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1 SS657a Midterm Exam, October 7 th 008 pg. SS57a Midterm Exam Monday Oct 7 th 008, 6:30-9:30 PM Talbot College 34 and 343 You may use simple, non-programmable scientific calculators. This exam has 5 questions worth a total of 5 marks on 6 pages, including this cover page. Attempt All questions. Name: Student Number: For Marker s Use only: Good luck! :+) Question Mark Question Mark Total

2 SS657a Midterm Exam, October 7 th 008 pg. Q (6 marks): A poll conducted by an election candidate asked two questions: () Do you support the candidate s position on environmental issues and (): Do you support the candidate s position on income tax cuts? A total of 00 responses were received; 600 said yes to the first question and 400 said yes to the second. If 300 respondents said no to the environmental question and yes to the tax cut question, how many said yes to the environmental question but no to the tax cut question? A = yes to q, #(A) = 600 B = yes to q, #(B) = 400 c #( A B) 300 c #( AB) #( B) #( A B) c #( AB ) #( A) #( AB) So there are 500 people said yes to the environment question but no to the tax cut question.

3 SS657a Midterm Exam, October 7 th 008 pg. 3 Q (5 marks). Write down Kolmogorov s Axioms for characterizing the probability function P on a finite sample space S. P: for all A S, P( A) 0 P : P( S) P3: If A and B are mutually exclusive, ie A B, Then P( A B) P( A) P( B)

4 SS657a Midterm Exam, October 7 th 008 pg. 4 Q3 (6 marks) Three events: A, B, and C are defined on a sample space S. Given that P(A) = 0., P(B) = 0. and P(C) = 0.3, what is the smallest possible value for P[(A U B U C) C ]? If we want P[(A U B U C) C ] to be as small as possible then we want P[(A U B U C)] to be as big as possible. That means we don t want any overlap. ] Therefore, P[(A U B U C)] = = 0.6 And P[(A U B U C) C ] = 0.4

5 SS657a Midterm Exam, October 7 th 008 pg. 5 Q4 (6 marks) Two fair dice are rolled. What is the probability that the number on the first die was at least as large as 4 given that the sum of the two dice was 8? Possible outcomes: (, 6), (3, 5), (4, 4), (5, 3), (6, 6) 5 equally likely outcome have sum of two dice = 8, only 3 of the 5 have first die at least as large as 4. So P(first die 4\sum=8) is 3/5.

6 SS657a Midterm Exam, October 7 th 008 pg. 6 Q5 (0 marks): When Xiaohong wants to contact her boyfriend and she knows he is not at home, she is twice as likely to send him an as she is to leave a message on his answering machine. The probability that he responds to her within 3 hours is 80%; his chance of responding to a phone message within 3 hours is 90%. Suppose he responded to the message she left this morning within hours. What is the probability that Xiaohong was communicating with him via ? P( ) / 3 P( phone) / 3 P( response ) 0.8, P( response ) 0.9 B event she responded A event she sent him an A Event she him a phone message P( B \ A) P( A) P( A \ B) PB ( \ A ) P( A ) P( B \ A ) P( A ) (0.8)( / 3) % (0,8)( / 3) (0.9)(/ 3) Therefore, the probability that she contact him visa was 64% [make sense: This is slightly less than base rate of /3 bince phone is more successful response setting method].

7 SS657a Midterm Exam, October 7 th 008 pg. 7 Q6 (6 marks): A desk has four drawers. Drawer contains two gold coins, Drawers and 3 each contain two silver coins, and Drawer 4 contains one silver coin and one gold coin. A coin is drawn at random from a drawer selected at random. Suppose the coin selected was silver. What is the probability that the other coin in the drawer was gold? P( A \ B) 4 B A i event coin drawn was silver event drawer i was picked Only was for other coin to be gold is if drawer 4 was picked. P( B \ A4) P( A4) P( B \ A ) P( A ) P( B \ A ) P( A ) P( B \ A ) P( A ) P( B \ A ) P( A ) ( )( ) (0)( ) ()( ) ()( ) ( )( ) Check: 5 silver coins could be picked only one of them is with a gold coin.

8 SS657a Midterm Exam, October 7 th 008 pg. 8 Q7 (0 marks): A computer is instructed to generate a random sequence using the digits 0 through 9; repetitions are permissible. What is the shortest length the sequence can be and still have at least a 70% probability of containing at least one 4? P(at least 4) = - P( no 4s) Find smallest N each digit is 4 with with probability, 0 9 or not 4 with probability. 0 With N digits N (0.) (0.9) 0 (0.9) N 0 N 0 N N (0.9) 0.7 or (0.9) 0.3 N

9 SS657a Midterm Exam, October 7 th 008 pg. 9 Q8 (0 marks) Experience has shown that only /3 of all patients having a certain disease will recover if given the standard treatment. A new drug is to be tested on a group of volunteers. If Health Canada, which regulates new drugs, requires that at least 7 of these patients recover before it will licence the new drug, what is the probability that the treatment will be rejected even though it increases the recovery rate to ½? 0 6 P( recover) P( recover) P( recover) = P ( P) P ( P) P ( P) 0 6 ( here P P ) % 4096

10 SS657a Midterm Exam, October 7 th 008 pg. 0 Q9 (6 marks): Suppose that f(x) is the probability distribution function corresponding to a continuous random variable which takes values only in the interval [,3]. Which of the following sketches of f(x) cannot be valid? Why? Check areas covered in[,3]: a) Area =, not valid b) Area is not possible to be, not valid c) Area, not valid d) Area, valid

11 SS657a Midterm Exam, October 7 th 008 pg. Q0 (0 marks): Suppose that the random variable X has probability density function f(x) = K, c x d, with c < d and for K > 0. a) In terms of c and d, what is the value of K? k d c b) Find E[X] d d d c xdx x c ( d c ) d c d c ( d c) c) Find Var[X] c d x dx d c d c c 3 d d c x c d c 3 d c d cd c 3( d c) d cd c d cd c 3 4 d cd c ( d c ) 6

12 SS657a Midterm Exam, October 7 th 008 pg. Q (0 marks) Persons suffering from purple spot fever have purple spots on their face for a period of time Y, where Y is a random variable described by the continuous probability density function f Y (y) = 3y /8, 0 y, (f Y (y) = 0, otherwise) where Y is measured in weeks. What is the probability that Maxwell, who began suffering from purple spot fever today, will still have purple spots week from now, when his class photos are taken? 0 3y dy 8 3 y

13 SS657a Midterm Exam, October 7 th 008 pg. 3 Q (0 marks) If f Y (y) = y/k, 0 y k, for what value of k does Var(Y) =? k 0 y y k k 0 dy 0 k k k E Y k 3 k dy 0 k k 0 y y k 3 3 k 3 4 y y k k dy 0 k k 0 E Y Var Y k k k 4k k k Var Y k k

14 SS657a Midterm Exam, October 7 th 008 pg. 4 Q3 (0 marks) Suppose that the random variable Y has probability density function f Y (y) = /y 3, y, (f Y (y) = 0, y < ) a) Show that this is a valid probability density function. 3 dy y y b) Find the expected value of Y. y dy dy y y y 3 c) However, show that Y does not have a finite variance. y Var( y) E( y ) ( E( y)) dy dy 3 Not finite lim R R dy lim ln R y R y y

15 SS657a Midterm Exam, October 7 th 008 pg. 5 Q4 (0 marks): Assume that the number of hits X that a baseball teams makes in an inning of a baseball game has a Poisson distribution. If the probability that a team makes zero hits in an inning is /3: a) what are their chances of getting two or more hits in the first inning? 0 e P( X 0) e ln 3 0! 3 e P( X ) e ln 3! 3 P( X 0) P( X ) ln 3 ln 3 ( ln 3) 3 3 ln 3 P( X ) P( X 0) P( X ) ( ln 3) 30.05% 3 3 b) What is the probability that they get 0 hits in a regular 9 inning game? game 9ln ln3 0 (9ln 3) 0 e (9ln 3) 3 (3ln 3) ln 3 P(0 Hits) % 0! 0! 0!

16 SS657a Midterm Exam, October 7 th 008 pg. 6 Q5 (0 marks) An experiment is performed independently 0 times. The results are,,0,0,0,0,,,0,. If the same experiment were performed again 5 times (independent of the above results), estimate the probability that exactly three of the five new experiments result in a? 5 successes out of 0 trials: Estimate P( success) % 3 3